40 pages + 23 pages Appendices, V2 added references, fixed typos; V3 added references, minor clarifications,V4 added new section on tensor and hypermultiplets, revtex format
Bianchi attractors are near horizon geometries with homogeneous symmetries in the spatial directions. We construct supersymmetric Bianchi attractors in $\mathcal{N}=2, d=4,5$ gauged supergravity. In $d=4$ we consider gauged supergravity coupled to vector and hypermultiplets. In $d=5$ we consider gauged supergravity coupled to vector multiplets with a generic gauging of symmetries of the scalar manifold and the $U(1)_R$ gauging of the $R$-symmetry. Analyzing the gaugino conditions we show that when the fermionic shifts do not vanish there are no supersymmetric Bianchi attractors. This is analogous to the known condition that for maximally supersymmetric solutions, all the fermionic shifts must vanish. When the central charge satisfies an extremization condition, some of the fermionic shifts vanish and supersymmetry requires that the symmetries of the scalar manifold do not be gauged. This allows supersymmetric Bianchi attractors sourced by massless gauge fields and a cosmological constant. In five dimensions in the Bianchi I class we show that the anisotropic $AdS_3\times\mathbb{R}^2$ solution is $1/2$ BPS. We also construct a new class of $1/2$ BPS Bianchi III geometries labeled by the central charge. When the central charge takes a special value the Bianchi III geometry reduces to the known $AdS_3\times\mathbb{H}^2$ solution. For the Bianchi V and VII classes the radial spinor breaks all of supersymmetry. We briefly discuss the conditions for possible massive supersymmetric Bianchi solutions by generalizing the matter content to include tensor, hypermultiplets and a generic gauging on the R symmetry.
Bianchi attractors are homogeneous but anisotropic extremal black brane horizons. We study the $AdS_3 \times \mathbb{H}^2$ solution which is a special case of Bianchi type III in a $U(1)_R$ gauged supergravity. For a wide range of values for certain free parameters in gauged supergravity, there exist a large class of solutions that satisfy conditions for the attractor mechanism to hold. We investigate the response of the solution against linearized fluctuations of the scalar field. The sufficient conditions for the attractor mechanism ensure that there exist a solution for the scalar fluctuati…
In this paper, we analyse the stability of extremal black brane horizons with homogeneous symmetry in the spatial directions in five dimensional gauged supergravity, under the fluctuations of the scalar fields about their attractor values. We examine the scalar fluctuation equations at the linearised level and demand that the fluctuations vanish as one approaches the horizon. Imposing certain restrictions on the Killing vectors used for gauging we find that the necessary conditions for stability are satisfied only by a subclass of the Bianchi metrics whose symmetry group factorises into a two …
In this paper we study generalized attractors in N=2 gauged supergravity theory in five dimensions coupled to arbitrary number of hyper, vector and tensor multiplets. We look for attractor solutions with constant anholonomy coefficients. By analyzing the equations of motion we derive the attractor potential. We further show that the generalized attractor potential can be obtained from the fermionic shifts. We study some simple examples and show that constant anholonomy gives rise to homogeneous black branes in five dimensions.
The attractor mechanism governs the near-horizon geometry of extremal black holes in ungauged 4D N=2 supergravity theories and in Calabi-Yau compactifications of string theory. In this paper, we study a natural generalization of this mechanism to solutions of arbitrary 4D N=2 gauged supergravities. We define generalized attractor points as solutions of an ansatz which reduces the Einstein, gauge field, and scalar equations of motion to algebraic equations. The simplest generalized attractor geometries are characterized by non-vanishing constant anholonomy coefficients in an orthonormal frame. …
We investigate the existence of supersymmetric static dyonic black holes with spherical horizon in the context of N= 2 U(1) gauged supergravity in four dimensions. We analyze the conditions for their existence and provide the general first-order flow equations driving the scalar fields and the metric warp factors from the asymptotic AdS4 geometry to the horizon. We work in a general duality-symmetric setup, which allows to describe both electric and magnetic gaugings. We also discuss the attractor mechanism and the issue of moduli (de-)stabilization.
We analyze BPS black hole attractors in 4d gauged supergravity in the presence of higher derivative supersymmetric terms, including a Weyl-squared-type action, and determine the resulting corrections to the Bekenstein-Hawking entropy. The near-horizon geometry AdS$_2 \times$S$^2$ (or other Riemann surface) preserves half of the supercharges in $N=2$ supergravity with Fayet-Iliopoulos gauging. We derive a relation between the entropy and the black hole charges that suggests via AdS/CFT how subleading corrections contribute to the supersymmetric index in the dual microscopic picture. Depending o…
Via a series of Kaluza-Klein (KK) and Scherk-Schwarz (SS) compactifications we relate BPS attractors and their complete (in general non-BPS) flows to a Minkowski vacuum in gauged supergravities with vanishing scalar potential in 4, 5, and 6 dimensions. This way we can look at a class of extremal non-BPS black holes and strings from IIB string theory viewpoint, keeping 4 supercharges on the horizon. Our results imply the existence of a dual 2d N = (0,2) superconformal field theory (SCFT) that originates from a parent N=(4,4) theory living on a D1-D5 system. This is achieved starting from the BP…
We relate across dimensions BPS attractors of black strings and black holes of various topology in gauged supergravities with nontrivial scalar potential. The attractors are of the form AdS$_{2, 3} \times Σ^{2, 3}$ in 4, 5, and 6 dimensions, and can be generalized to some higher dimensional analogs. Even though the attractor geometries admit standard Kaluza-Klein and Scherk-Schwarz reductions, their asymptotic AdS spaces in general do not. The resulting lower dimensional objects are black holes with runaway asymptotics in supergravity theories with no maximally symmetric vacua. Such classes of…
We perform a systematic search for static solutions in different sectors of 5d $N=8$ supergravities with compact and non-compact gauged R-symmetry groups, finding new and listing already known backgrounds. Due to the variety of possible gauge groups and resulting scalar potentials, the maximally symmetric vacua we encounter in these theories can be Minkowski, de Sitter, or anti-de Sitter. There exist BPS and non-BPS near-horizon geometries and full solutions with all these three types of asymptotics, corresponding to black holes, branes, strings, rings, and other black objects with more exotic…
We have investigated the attractor structure for the CMB fluctuations in composite inflation scenario within the gauged Nambu-Jona-Lasinio (NJL) model. Such composite inflation represents an attractor which can not be found in a fundamental scalar model. As is known, the number of inflationary models contains the attractor classified by the $α$-attractor model. It is found that the attractor inflation in the gauged NJL model corresponds to the $α= 2$ case.
We perform the generalised dimensional reduction of D=11 supergravity over three-dimensional group manifolds as classified by Bianchi. Thus, we construct eleven different maximal D=8 gauged supergravities, two of which have an additional parameter. One class of group manifolds (class B) leads to supergravities that are defined by a set of equations of motion that cannot be integrated to an action. All 1/2 BPS domain wall solutions are given. We also find a non-supersymmetric domain wall solution where the single transverse direction is time. This solution describes an expanding universe and up…
We construct classes of smooth metrics which interpolate from Bianchi attractor geometries of Types II, III, VI and IX in the IR to Lifshitz or $AdS_2 \times S^3$ geometries in the UV. While we do not obtain these metrics as solutions of Einstein gravity coupled to a simple matter field theory, we show that the matter sector stress-energy required to support these geometries (via the Einstein equations) does satisfy the weak, and therefore also the null, energy condition. Since Lifshitz or $AdS_2 \times S^3$ geometries can in turn be connected to $AdS_5$ spacetime, our results show that there …
We establish an attractor mechanism for the horizon metric of asymptotically locally AdS4 supersymmetric black holes. The Kähler moduli of the metric are not fixed in the asymptotic region, however supersymmetry dictates that the near horizon metric must be the constant curvature one. We show how this mechanism is realized in detail for four-dimensional N=2 gauged supergravity coupled to vector multiplets by focusing on the STU model. A similar analysis is performed for gauged supergravity theories in five, six, and seven dimensions where we establish the same mechanism by extending previous r…
Global N=2 supersymmetry in four dimensions with a gauged central charge is formulated in superspace. To find an irreducible representation of supersymmetry for the gauge connections a set of constraints is given. Then the Bianchi identities are solved subject to this set of constraints. It is shown that the gauge connection of the central charge is a N=2 vector multiplet. Moreover the Bogomol'nyi bound of the massive particle states is studied.
We consider the flow equations for $1/4$-BPS asymptotically AdS$_4$ static black holes in Fayet-Iliopoulos gauged supergravity, using very special geometry identities to obtain a simplified form in the most general case. Under mild assumptions on the form of the solution, we analyse the flow equations and find an explicit solution for arbitrary gauging and charge vectors within the chosen ansatz. Comparing with the corresponding attractor equations, we find that the solution is given in terms of exactly the same vector of parameters, implying that all regular attractors can be extended to full…
SU(2) gauge fields coupled to an axion field can acquire an isotropic background solution during inflation. We study homogeneous but anisotropic inflationary solutions in the presence of such (massless) gauge fields. A gauge field in the cosmological background may pose a threat to spatial isotropy. We show, however, that such models $\textit{generally}$ isotropize in Bianchi type-I geometry, and the isotropic solution is the attractor. Restricting the setup by adding an axial symmetry, we revisited the numerical analysis presented in Wolfson et.al (2020). We find that the reported numerical b…
Extremal black branes are of interest because they correspond to the ground states of field theories at finite charge density in gauge/gravity duality. The geometry of such a brane need not be translationally invariant in the spatial directions along which it extends. A less restrictive requirement is that of homogeneity, which still allows points along the spatial directions to be related to each other by symmetries. In this paper, we find large new classes of homogeneous but anisotropic extremal black brane horizons, which could naturally arise in gauge/gravity dual pairs. In 4+1 dimensional…
We consider four-dimensional $N=2$ supergravity coupled to vector- and hypermultiplets, where abelian isometries of the quaternionic Kähler hypermultiplet scalar manifold are gauged. Using the recipe given by Meessen and Ort\'ın in arXiv:1204.0493, we analytically construct a supersymmetric black hole solution for the case of just one vector multiplet with prepotential ${\cal F}=-iχ^0χ^1$, and the universal hypermultiplet. This solution has a running dilaton, and it interpolates between $\text{AdS}_2\times\text{H}^2$ at the horizon and a hyperscaling-violating type geometry at infinity, confor…
We apply Sen's entropy formalism to the study of the near horizon geometry and the entropy of asymptotically AdS black holes in gauged supergravities. In particular, we consider non-supersymmetric electrically charged black holes with AdS_2 xS^{d-2} horizons in U(1)^4 and U(1)^3 gauged supergravities in d=4 and d=5 dimensions, respectively. We study several cases including static/rotating, BPS and non-BPS black holes in Einstein as well as in Gauss-Bonnet gravity. In all examples, the near horizon geometry and black hole entropy are derived by extremizing the entropy function and are given ent…
Dynamics of the Bianchi~V cosmological model inspired by quintessential $α$-attractors
Universidad Católica del Norte（北天主教学会大学）；Durban University of Technology（德班理工大学）；North-West University（西北大学）；National Institute for Theoretical and Computational Sciences (NITheCS)（国家理论与计算科学研究所）
We study tilted perfect fluid cosmological models with a constant equation of state parameter in spatially homogeneous models of Bianchi type VI_h using dynamical systems methods and numerical experimentation, with an emphasis on their future asymptotic evolution. We determine all of the equilibrium points of the type VI_h state space (which correspond to exact self-similar solutions of the Einstein equations, some of which are new), and their stability is investigated. We find that there are vacuum plane-wave solutions that act as future attractors. In the parameter space, a `loophole' is sho…
We find a family of AdS$_2\times \mathcal{M}_4$ supersymmetric solutions of the six-dimensional $\mathrm{F}(4)$ gauged supergravity coupled to one vector multiplet that arises as a low energy description of massive type IIA supergravity on (warped) AdS$_6\times S^4$. $\mathcal{M}_4$ is either a Kähler-Einstein manifold or a product of two Riemann surfaces with a constant curvature metric. These solutions correspond to the near-horizon region of a family of static magnetically charged black holes. In the case where $\mathcal{M}_4$ is a product of Riemann surfaces, we successfully compare their …
Performing a Scherk-Schwarz dimensional reduction of D=11 supergravity on a three-dimensional group manifold we construct five D=8 gauged maximal supergravities whose gauge groups are the three-dimensional (non-)compact subgroups of SL(3,R). These cases include the Salam-Sezgin SO(3) gauged supergravity. We construct the most general half-supersymmetric domain wall solutions to these five gauged supergravities. The generic form is a triple domain wall solution whose truncations lead to double and single domain wall solutions. We find that one of the single domain wall solutions has zero potent…
In this paper we propose the identification in AdS4 N = 2 gauged supergravity of the coefficient $τ_{RR}$ of 3D N = 2 SCFTs. We constraint the structure of this function in supergravity by combining the results from unitarity, holography and localization. We show that our conjectured function is minimized by the exact R-charge, corresponding to a gravitational attractor for the scalars in the special geometry. We identify this mechanism with the supergravity dual of the $τ_{RR}$-minimization. We check this proposal in the ABJM model, comparing with the expectations from localization and the Ad…
We construct explicitly time-dependent exact solutions to the field equations of 6D gauged chiral supergravity, compactified to 4D in the presence of up to two 3-branes situated within the extra dimensions. The solutions we find are scaling solutions, and are plausibly attractors which represent the late-time evolution of a broad class of initial conditions. By matching their near-brane boundary conditions to physical brane properties we argue that these solutions (together with the known maximally-symmetric solutions and a new class of non-Lorentz-invariant static solutions, which we also pre…
Using the general recipe given in arXiv:0804.0009, where all timelike supersymmetric solutions of N=2, D=4 gauged supergravity coupled to abelian vector multiplets were classified, we construct the first examples of genuine supersymmetric black holes in AdS_4 with nonconstant scalar fields. This is done for various choices of the prepotential, amongst others for the STU model. These solutions permit to study the BPS attractor flow in AdS. We also determine the most general supersymmetric static near-horizon geometry and obtain the attractor equations in gauged supergravity. As a general featur…
We construct the so far unknown Lagrangian of D=6, N=2 F(4) Supergravity coupled to an arbitrary number of vector multiplets whose scalars span the coset manifold SO(4,n)/{SO(4) x SO(n)}. This is done first in the ungauged case and then extended to the compact gauging of SU(2) x G, where SU(2) is the R-symmetry diagonal subgroup of SU(2)_L x SU(2)_R = SO(4) and G is a compact subgroup of SO(n), n being the number of vector multiplets, and such that dim G = n. The knowledge of the Lagrangian allows in principle to refine the AdS_6/CFT_5 correspondence already discussed, as far as supersymmetric…
We establish general properties of supersymmetric flow equations and of the superpotential of five-dimensional N = 2 gauged supergravity coupled to vector and hypermultiplets. We provide necessary and sufficient conditions for BPS domain walls and find a set of algebraic attractor equations for N = 2 critical points. As an example we describe in detail the gauging of the universal hypermultiplet and a vector multiplet. We study a two-parameter family of superpotentials with supersymmetric AdS critical points and we find, in particular, an N = 2 embedding for the UV-IR solution of Freedman, Gub…
We analyze BPS black hole attractors in the conformal 4d gauged supergravity formalism and apply the technique known as supergravity localization in order to evaluate Sen's quantum entropy function in the $\mathrm{AdS}_2\!\times\!\mathrm{S}^2$ near-horizon geometry. Under certain assumptions, we reduce the exact expression of the functional integral to a finite-dimensional integral for a number of supersymmetric black holes in gauged supergravity with AdS asymptotics subject to a holographic description via a dual field theory. Examples include the asymptotically $\mathrm{AdS}_4\!\times\!\math…
In this paper we address the question how to discriminate whether the gauged isometry group G_Sigma of the Kahler manifold Sigma that produces a D-type inflaton potential in a Minimal Supergravity Model is elliptic, hyperbolic or parabolic. We show that the classification of isometries of symmetric cosets can be extended to non symmetric Sigma.s if these manifolds satisfy additional mathematical restrictions. The classification criteria established in the mathematical literature are coherent with simple criteria formulated in terms of the asymptotic behavior of the Kahler potential K(C) = 2 J(…
We investigate the attractor mechanism in gauged supergravity in the presence of higher derivatives terms. In particular, we discuss the attractor behaviour of static black hole horizons in anti-de Sitter spacetime by using the effective potential approach as well as Sen's entropy function formalism. We use the holographic techniques to interpret the moduli flow as an RG flow towards the IR attractor horizon. We find that the holographic c-function obeys the expected properties and point out some subtleties in understanding attractors in AdS.
We study supersymmetric black holes in $AdS_4$ in the framework of four dimensional gauged $\N=2$ supergravity coupled to hypermultiplets. We derive the flow equations for a general electrically gauged theory where the gauge group is Abelian and, restricting them to the fixed points, we derive the gauged supergravity analogue of the attractor equations for theories coupled to hypermultiplets. The particular models we analyze are consistent truncations of M-theory on certain Sasaki-Einstein seven-manifolds. We study the space of horizon solutions of the form $AdS_2\times Σ_g$ with both electric…
Continuing previous work, we show the existence of stable, anisotropic future attractors in Bianchi invariant sets with a $p$-form field ($p\,\in\,\{1,3\}$) and a perfect fluid. In particular, we consider the not previously investigated Bianchi invariant sets $\mathcal{B}$(II), $\mathcal{B}$(IV), $\mathcal{B}$(VII$_0$) and $\mathcal{B}$(VII$_{h})$ and examine their asymptotic behaviour. We find that the isolated equilibrium set Wonderland is a future attractor on all of its existence ($2/3<\,γ\,<2$) in all these sets except in $\mathcal{B}$(II), where the peculiar equilibrium sets Edge and Rop…
We investigate extremal Reissner-Nordstrom-AdS black holes in four-dimensional N=2 abelian gauged supergravity. We find a new attractor equation which is not reduced to the one in the asymptotically flat spacetime. Focusing on so-called the T^3-model with a single neutral vector multiplet, we obtain non-supersymmetric extremal Reissner-Nordstrom-AdS black hole solutions with regular event horizon in the D0-D4 and the D2-D6 charge configurations. The negative cosmological constant emerges even without the Fayet-Iliopoulos parameters. Furthermore, we also argue the formal description of the non-…
We consider black holes in five-dimensional $N=2$ $\text{U}(1)$-gauged supergravity coupled to vector multiplets, with horizons that are homogeneous but not isotropic. We write down the equations of motion for electric and magnetic ansätze, and solve them explicitely for the case of pure gauged supergravity with magnetic $\text{U}(1)$ field strength and Sol horizon. The thermodynamics of the resulting solution, which exhibits anisotropic scaling, is discussed. If the horizon is compactified, the geometry approaches asymptotically a torus bundle over AdS$_3$. Furthermore, we prove a no-go theor…
This thesis is focused on supersymmetric objects in gauged supergravities and their interpretation in string theory. Three particular theories are considered and analyzed: matter-coupled $d=4, 5$ abelian gauged supergravities with 8 supercharges and minimal $d=7$ gauged supergravity with 16 supercharges. Respectively in the $d=4$ and $d=5$ cases, the BPS first-order flows of static extremal black holes and black strings are studied using the Hamilton-Jacobi formalism. The attractor mechanism is formulated for the case of couplings to hypermultiplets and two exact BPS black hole solutions are d…
We construct four-dimensional domain wall solutions of N=2 gauged supergravity coupled to vector and to hypermultiplets. The gauged supergravity theories that we consider are obtained by performing two types of Abelian gauging. In both cases we find that the behaviour of the scalar fields belonging to the vector multiplets is governed by the so-called attractor equations known from the study of BPS black hole solutions in ungauged N=2 supergravity theories. The scalar fields belonging to the hypermultiplets, on the other hand, are either constant or exhibit a run-away behaviour. These domain w…
We consider a deformation of the well-known stu model of $N=2$, $D=4$ supergravity, characterized by a non-homogeneous special Kähler manifold, and by the smallest electric-magnetic duality Lie algebra consistent with its upliftability to five dimensions. We explicitly solve the BPS attractor equations and construct static supersymmetric black holes with radial symmetry, in the context of $\text{U}(1)$ dyonic Fayet-Iliopoulos gauging, focussing on axion-free solutions. Due to non-homogeneity of the scalar manifold, the model evades the analysis recently given in the literature. The relevant ph…
We derive both BPS and non-BPS first-order flow equations for magnetically charged black strings in five-dimensional N=2 abelian gauged supergravity, using the Hamilton-Jacobi formalism. This is first done for the coupling to vector multiplets only and U(1) Fayet-Iliopoulos (FI) gauging, and then generalized to the case where also hypermultiplets are present, and abelian symmetries of the quaternionic hyperscalar target space are gauged. We then use these results to derive the attractor equations for near-horizon geometries of extremal black strings, and solve them explicitely for the case whe…
We consider N=2 supergravity in four dimensions, coupled to an arbitrary number of vector- and hypermultiplets, where abelian isometries of the quaternionic hyperscalar target manifold are gauged. Using a static and spherically or hyperbolically symmetric ansatz for the fields, a one-dimensional effective action is derived whose variation yields all the equations of motion. By imposing a sort of Dirac charge quantization condition, one can express the complete scalar potential in terms of a superpotential and write the action as a sum of squares. This leads to first-order flow equations, that …
We construct new models of inflation and spontaneous supersymmetry breaking in de Sitter vacuum, with a single chiral superfield, where inflaton is the superpartner of the goldstino. Our approach is based on hyperbolic Kähler geometry, and a gauged (non-axionic) $U(1)_R$ symmetry rotating the chiral scalar field by a phase. The $U(1)_R$ gauge field combines with the angular component of the chiral scalar to form a massive vector, and single-field inflation is driven by the radial part of the scalar. We find that in a certain parameter range they can be approximated by simplest Starobinsky-like…
We elaborate further on the static supersymmetric AdS_4 black holes found in arXiv:0911.4926, investigating thoroughly the BPS constraints for spherical symmetry in N = 2 gauged supergravity in the presence of Fayet-Iliopoulos terms. We find Killing spinors that preserve two of the original eight supercharges and investigate the conditions for genuine black holes free of naked singularities. The existence of a horizon is intimately related with the requirement that the scalars are not constant, but given in terms of harmonic functions in analogy to the attractor flow in ungauged supergravity. …
We present a new consistent truncation of $D=11$ supergravity to $D=4$ $\mathcal{N}=2$ minimal gauged supergravity, on the seven-dimensional internal Riemannian space corresponding to the most general class of $D=11$ solutions with an AdS$_{4}$ factor and $\mathcal{N}=2$ supersymmetry. A truncation ansatz is proposed and its consistency checked at the level of the $D=11$ Bianchi identity, bosonic equations of motion, and supersymmetry variations of the gravitino. The general class includes an $\mathcal{N}=2$ AdS$_{4}$ solution dual to the conformal, low-energy physics phase corresponding to a …
